If the behaviour of grammars for strings is directly used in the two-dimensional case, we have to overcome the problem of 'shearing'. 'Shearing' means replacing one sub-picture with another picture of different size. The matrix grammars avoid this problem by first generating horizontally and then generating vertically afterwards. This section deals with the isometric array grammars from A. Rosenfeld \cite{rosenfeldpicture}, which use another attempt. Here, we can define rules like in the one-dimensional case. Nevertheless, it is required that the sizes of the left and the right side of a rule are equal. To give grammars the possibility to enlarge the size of a picture, the start symbol is surrounded by an infinite number of blank symbols. These blank symbols can be replaced by the rules of the grammar. 

\begin{definition}
	A quintuple $G = (V, V_T, \Pi, S, \#)$ is called an \emph{isometric array grammar} (IAG) where 
	\begin{compactitem}
		\item $V$ is a non-empty finite set of working vocabulary,
		\item $V_T$ is a non-empty finite set of terminal symbols ($V_T \subset V$),
		\item $\Pi$ is a set of production rules, where for every rule $(A \rightarrow B) \in \Pi$ $l_1(A) = l_1(B)$ and $l_2(A) = l_2(B)$. Furthermore, any side of a rule must contain at least one non-blank symbol,
		\item $S$ is the start symbol, and
		\item $\#$ is the blank symbol. 
	\end{compactitem}
\end{definition}

Firstly, derivations for IAG have to be defined. $p' \in V^{*, *}$ derive directly from $p \in V^{*, *}$ ($p \Rightarrow p'$), if $p$ and $p'$ are identical except to position $(i, j)$, ($1 \leq i \leq l_1(p), 1 \leq j \leq l_2(p)$) where $p$ consists of a sub-picture $q$ and $p'$ consists of a sub-picture $q'$ of the same size, and $(q \rightarrow q') \in \Pi$ is a rule. As usual the reflexive, transitive closure of $\Rightarrow$ is denoted as $\overset{*}{\Rightarrow}$. 

\begin{definition}
	The set of pictures generated by an IAG G is 
	
	\[L(G) = \{p \mid S \overset{*}{\Rightarrow} p\}\]
	
	and is called an \emph{isometric array language} (IAL). 
\end{definition}

To better understand the generative power of IAG's, we continue with an example from~\cite{morita2004twodimensional}. 

\begin{example}
	Let $G = (V, V_T, \Pi, S, \#)$ be an isometric array grammar with
	\begin{compactitem}
		\item $V = \{S, H, V, a, \#\}$ is the working alphabet,
		\item $V_T = \{a\}$ is the set of terminal symbols and
		\item $\Pi$ contains the following rules:
		\begin{compactitem}
			%rule 1
			\item \boxed{
					\begin{aligned}
						\begin{matrix}
							S & \#
						\end{matrix}
					\end{aligned}
				}
				$\rightarrow$
				\boxed{
					\begin{aligned}
						\begin{matrix}
							a & S
						\end{matrix}
					\end{aligned}
				}
			%rule 2
			\item \boxed{
					\begin{aligned}
						\begin{matrix}
							S & \# & \# \\[-0.5ex]
			 				& \# &
						\end{matrix}
					\end{aligned}
				}
				$\rightarrow$
				\boxed{
					\begin{aligned}
						\begin{matrix}
							a & a & H \\[-0.5ex]
			 				& V &
						\end{matrix}
					\end{aligned}
				}
			%rule 3
			\item \boxed{
					\begin{aligned}
						\begin{matrix}
							H & \#
						\end{matrix}
					\end{aligned}
				}
				$\rightarrow$
				\boxed{
					\begin{aligned}
						\begin{matrix}
							a & H
						\end{matrix}
					\end{aligned}
				}
			%rule 4
			\item \boxed{
					\begin{aligned}
						\begin{matrix}
							H
						\end{matrix}
					\end{aligned}
				}
				$\rightarrow$
				\boxed{
					\begin{aligned}
						\begin{matrix}
							a
						\end{matrix}
					\end{aligned}
				}
			%rule 5
			\item \boxed{
					\begin{aligned}
						\begin{matrix}
							V \\[-0.5ex]
			 				\#
						\end{matrix}
					\end{aligned}
				}
				$\rightarrow$
				\boxed{
					\begin{aligned}
						\begin{matrix}
							a \\[-0.5ex]
			 				V
						\end{matrix}
					\end{aligned}
				}
			%rule 6
			\item \boxed{
					\begin{aligned}
						\begin{matrix}
							V
						\end{matrix}
					\end{aligned}
				}
				$\rightarrow$
				\boxed{
					\begin{aligned}
						\begin{matrix}
							a
						\end{matrix}
					\end{aligned}
				}. 
		\end{compactitem}
	\end{compactitem}
	
	When starting with S, a string of a's is generated first. At a random point, the generation splits into a horizontal and a vertical part. From this point, a vertical and a horizontal string of a's can be generated. Thus, the language generated by G contains pictures of T-shaped a's. 
\end{example}

An example picture of size (3, 4) can be derived as follows: 

\[
\begin{aligned}
\begin{matrix}
\# & \# & \# & \# & \# \\[-0.5ex]
\# & S & \# & \# & \# \\[-0.5ex]
\# & \# & \# & \# & \# \\[-0.5ex]
\# & \# & \# & \# & \# \\[-0.5ex]
\# & \# & \# & \# & \# 
\end{matrix}
\end{aligned}
\Rightarrow
\begin{aligned}
\begin{matrix}
\# & \# & \# & \# & \# \\[-0.5ex]
\# & a & a & H & \# \\[-0.5ex]
\# & \# & V & \# & \# \\[-0.5ex]
\# & \# & \# & \# & \# \\[-0.5ex]
\# & \# & \# & \# & \# 
\end{matrix}
\end{aligned}
\Rightarrow
\begin{aligned}
\begin{matrix}
\# & \# & \# & \# & \# \\[-0.5ex]
\# & a & a & a & \# \\[-0.5ex]
\# & \# & V & \# & \# \\[-0.5ex]
\# & \# & \# & \# & \# \\[-0.5ex]
\# & \# & \# & \# & \# 
\end{matrix}
\end{aligned}
\]

\[
\Rightarrow
\begin{aligned}
\begin{matrix}
\# & \# & \# & \# & \# \\[-0.5ex]
\# & a & a & a & \# \\[-0.5ex]
\# & \# & a & \# & \# \\[-0.5ex]
\# & \# & V & \# & \# \\[-0.5ex]
\# & \# & \# & \# & \# 
\end{matrix}
\end{aligned}
\Rightarrow
\begin{aligned}
\begin{matrix}
\# & \# & \# & \# & \# \\[-0.5ex]
\# & a & a & a & \# \\[-0.5ex]
\# & \# & a & \# & \# \\[-0.5ex]
\# & \# & a & \# & \# \\[-0.5ex]
\# & \# & \# & \# & \# 
\end{matrix}
\end{aligned}
\]

\cite{morita2004twodimensional} proposed some restrictions of the rules to induce a Chomsky-like hierarchy:

\begin{definition}
	Let $G = (V, V_T, \Pi, S, \#)$ be an isometric array grammar. We call G a
	\begin{compactitem}
		\item \emph{monotonic array grammar} (MAG), if in any rule non-$\#$'s are never rewritten by $\#$'s, 
		\item \emph{context-free array grammar} (CFAG), if for any rule $(A \rightarrow B) \in \Pi$, A consists of exactly one terminal (with $\#$'s around), 
		\item \emph{regular array grammar} (RAG), if for any rule $(A \rightarrow B) \in \Pi$, A consists of exactly one terminal (with possibly one $\#$) and B consists of exactly one terminal and one or zero non-terminals. 
	\end{compactitem}
\end{definition}

The corresponding languages to MAG, CFAG and RAG are called \emph{monotonic array language} (MAL), \emph{context-free array language} (CFAL) and \emph{regular array language} (RAL). The families of languages generated by IAG's, MAG's, CFAG's and RAG's are denoted by $\mathscr{L}(IAL)$, $\mathscr{L}(MAL)$, $\mathscr{L}(CFAL)$ and $\mathscr{L}(RAL)$. 

From~\cite{morita2004twodimensional} we obtain the hierarchy: $\mathscr{L}(RAL) \subset \mathscr{L}(CFAL) \subset \mathscr{L}(MAL) \subset \mathscr{L}(IAL)$. 

Conclusively, it can be stated that both grammar approaches create hierarchies of two-dimensional languages. These hierarchies are quite similar to the one-dimensional case. Whereas for matrix grammars there are many extensions and further developments, the isometric array grammars are not examined that well, but there is an attempt to study what happens if several rules are applied simultaneously \cite{rosenfeldpicture}. 

Before we are going to determine which candidates can be used as regular two-dimensional languages, we look at the possibility to accept two-dimensional languages by automata. 